All of the one-digit day-numbers, 1-9, when reversed, are themselves, so have the property. That’s 9.

All of the two-digit day-numbers, 10-99, when reversed, are also day-numbers, except that those ending in 0 aren’t allowed (i.e., their reversal isn’t). That’s another 81.

It gets more complicated with the three-digit day-numbers.

Let’s start by excluding the day-numbers that end in 0, or 4 or larger. This leaves the following:

10x – 36x, where x is 1, 2 or 3

All of these when reversed are also day-numbers. That’s another 27*3 = 81. So we have 171 overall.

It’s a nice coincidence that there are 81 two-digit numbers and 81 three-digit numbers in the set. If any three-digit number were allowed, not just day-numbers between 1 and 366, then there would be 810 three-digit numbers in the set, or 900 overall. The number of possibilities for a fixed number of digits roughly increases with the square of the upper bound.